Elliptic curves with large Tate-Shafarevich groups over $\mathbb{F}_q(t)$

Richard Griffon; Guus de Wit

Corpus ID: 198985822

Elliptic curves with large Tate-Shafarevich groups over $\mathbb{F}_q(t)$

@article{Griffon2019EllipticCW,
  title={Elliptic curves with large Tate-Shafarevich groups over \$\mathbb\{F\}_q(t)\$},
  author={Richard Griffon and Guus de Wit},
  journal={arXiv: Number Theory},
  year={2019}
}

Richard Griffon, Guus de Wit

Published 2019

Mathematics

arXiv: Number Theory

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$. The… CONTINUE READING

View PDF on arXiv